cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A133070 a(n) = n^5 - n^3 - n^2.

Original entry on oeis.org

0, -1, 20, 207, 944, 2975, 7524, 16415, 32192, 58239, 98900, 159599, 246960, 368927, 534884, 755775, 1044224, 1414655, 1883412, 2468879, 3191600, 4074399, 5142500, 6423647, 7948224, 9749375, 11863124, 14328495, 17187632, 20485919, 24272100, 28598399, 33520640, 39098367, 45394964
Offset: 0

Views

Author

Omar E. Pol, Nov 01 2007

Keywords

Comments

Exponents are prime numbers in decreasing order.

Examples

			a(7)=16415 because 7^5=16807, 7^3=343, 7^2=49 and we can write 16807-343-49=16415.

Links

Crossrefs

Cf. A000290, A000578, A000584, A011379, A045991, A100019.
Cf. A133071, A133072, A133073.

Programs

  • Magma
    [n^5-n^3-n^2: n in [0..50]]; // Vincenzo Librandi, Dec 15 2010
  • Mathematica
    Table[n^5-n^3-n^2,{n,0,40}] (* or *) LinearRecurrence[ {6,-15,20,-15,6,-1},{0,-1,20,207,944,2975},41] (* Harvey P. Dale, Jul 23 2011 *)
  • PARI
    for(n=0,50, print1(n^5 - n^3 - n^2, ", ")) \\ G. C. Greubel, Oct 20 2017

Formula

a(n) = n^5 - n^3 - n^2.
G.f.: x*(-1 +26*x + 72*x^2 + 22*x^3 + x^4)/(1-x)^6. - R. J. Mathar, Nov 14 2007
a(n) = 6*a(n-1) -15*a(n-2) +20*a(n-3) -15*a(n-4) +6*a(n-5) -a(n-6), with a(0)=0, a(1)=-1, a(2)=20, a(3)=207, a(4)=944, a(5)=2975. - Harvey P. Dale, Jul 23 2011

Extensions

More terms from Vincenzo Librandi, Dec 15 2010

A133072 a(n) = n^5 + n^3 - n^2.

Original entry on oeis.org

0, 1, 36, 261, 1072, 3225, 7956, 17101, 33216, 59697, 100900, 162261, 250416, 373321, 540372, 762525, 1052416, 1424481, 1895076, 2482597, 3207600, 4092921, 5163796, 6447981, 7975872, 9780625, 11898276, 14367861, 17231536, 20534697, 24326100, 28657981, 33586176, 39170241, 45473572
Offset: 0

Views

Author

Omar E. Pol, Nov 01 2007

Keywords

Comments

Exponents are the prime numbers in decreasing order.

Examples

			a(7)=17101 because 7^5=16807, 7^3=343, 7^2=49 and we can write 16807+343-49=17101.

Links

Crossrefs

Cf. A000290, A000578, A000584, A011379, A045991, A100019.
Cf. A133070, A133071, A133073.

Programs

  • Magma
    [n^5+n^3-n^2: n in [0..50]]; // Vincenzo Librandi, Dec 15 2010
  • Mathematica
    Table[n^5 + n^3 - n^2, {n, 0, 50}] (* G. C. Greubel, Oct 20 2017 *)
  • PARI
    for(n=0,50, print1(n^5 + n^3 - n^2, ", ")) \\ G. C. Greubel, Oct 20 2017

Formula

a(n) = n^5 + n^3 - n^2.
G.f.: x*(1 + 30*x + 60*x^2 + 26*x^3 + 3*x^4)/(1-x)^6. - R. J. Mathar, Nov 14 2007

Extensions

More terms from Vincenzo Librandi, Dec 15 2010

A133073 a(n) = n^5 + n^3 + n^2.

Original entry on oeis.org

0, 3, 44, 279, 1104, 3275, 8028, 17199, 33344, 59859, 101100, 162503, 250704, 373659, 540764, 762975, 1052928, 1425059, 1895724, 2483319, 3208400, 4093803, 5164764, 6449039, 7977024, 9781875, 11899628, 14369319, 17233104, 20536379, 24327900, 28659903, 33588224, 39172419, 45475884
Offset: 0

Views

Author

Omar E. Pol, Nov 01 2007

Keywords

Comments

Exponents are prime numbers in decreasing order.

Examples

			a(7) = 17199 because 7^5 = 16807, 7^3 = 343, 7^2 = 49 and we can write 16807 + 343 + 49 = 17199.

Links

Crossrefs

Cf. A000290, A000578, A000584, A011379, A045991, A100019.
Cf. A133070, A133071, A133072.

Programs

  • Magma
    [n^5+n^3+n^2: n in [0..50]]; // Vincenzo Librandi, Dec 15 2010
  • Mathematica
    Total[#^{5,3,2}]&/@Range[0,40]  (* Harvey P. Dale, Jan 18 2011 *)
LinearRecurrence[{6,-15,20,-15,6,-1},{0,3,44,279,1104,3275},35] (* James C. McMahon, Mar 10 2025 *)
  • PARI
    for(n=0,50, print1(n^5 + n^3 + n^2, ", ")) \\ G. C. Greubel, Oct 20 2017

Formula

G.f.: x*(3 + 26*x + 60*x^2 + 30*x^3 + x^4)/(1-x)^6. - R. J. Mathar, Nov 14 2007
a(n) = n^2*(n^3 + n + 1). - Wesley Ivan Hurt, Mar 02 2023

Extensions

More terms from Vincenzo Librandi, Dec 15 2010

A134631 a(n) = 5*n^5 - 3*n^3 + 2*n^2. Coefficients and exponents are the prime numbers in decreasing order.

Original entry on oeis.org

0, 4, 144, 1152, 4960, 15300, 38304, 83104, 162432, 293220, 497200, 801504, 1239264, 1850212, 2681280, 3787200, 5231104, 7085124, 9430992, 12360640, 15976800, 20393604, 25737184, 32146272, 39772800, 48782500, 59355504, 71686944, 85987552, 102484260, 121420800, 143058304, 167675904, 195571332, 227061520
Offset: 0

Views

Author

Omar E. Pol, Nov 04 2007

Keywords

Examples

			a(4)=4960 because 4^5=1024, 5*1024=5120, 4^3=64, 3*64=192, 4^2=16, 2*16=32 and we can write 5120 - 192 + 32 = 4960.

Links

Crossrefs

Cf. A000290, A000578, A000584, A045991, A100019, A133071.

Programs

  • Magma
    [5*n^5-3*n^3+2*n^2: n in [0..50]]; // Vincenzo Librandi, Dec 14 2010
  • Mathematica
    Table[5n^5-3n^3+2n^2,{n,0,40}] (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{0,4,144,1152,4960,15300},40] (* Harvey P. Dale, Jan 20 2023 *)

Formula

a(n) = 5*n^5 - 3*n^3 + 2*n^2.
G.f.: 4x*(1+30x+87x^2+32x^3)/(1-x)^6. - R. J. Mathar, Nov 14 2007

Extensions

More terms from Vincenzo Librandi, Dec 14 2010

A133062 a(n) = 5*p^5 - 3*p^3 + 2*p^2 with p = prime(n).

Original entry on oeis.org

144, 1152, 15300, 83104, 801504, 1850212, 7085124, 12360640, 32146272, 102484260, 143058304, 346570564, 579077604, 734807392, 1146417984, 2090536452, 3574012320, 4222308004, 6749732224, 9020083104, 10364201572, 15383815360, 19693501632, 27918198180, 42933982084, 52547432004
Offset: 1

Views

Author

Omar E. Pol, Nov 05 2007

Keywords

Examples

			a(4)=83104 because the 4th prime is 7, 5*7^5=84035, 3*7^3=1029, 2*7^2=98 and we can write 84035 - 1029 + 98 = 83104.

Crossrefs

Cf. A045991, A133071.

Programs

  • Magma
    [5*p^5-3*p^3+2*p^2: p in PrimesUpTo(200)] // Vincenzo Librandi, Dec 15 2010
  • Mathematica
    5#^5-3#^3+2#^2&/@Prime[Range[30]] (* Harvey P. Dale, Mar 07 2017 *)

Formula

a(n) = 5*A050997(n) - 3*A030078(n) + 2*A001248(n) = A134631(A000040(n)).

Extensions

More terms from Vincenzo Librandi, Dec 15 2010
Showing 1-5 of 5 results.

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